Geometry & Topology

Hyperbolic jigsaws and families of pseudomodular groups, I

Beicheng Lou, Ser Peow Tan, and Anh Duc Vo

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Abstract

We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.

Article information

Source
Geom. Topol., Volume 22, Number 4 (2018), 2339-2366.

Dates
Received: 15 November 2016
Revised: 23 June 2017
Accepted: 9 October 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584824

Digital Object Identifier
doi:10.2140/gt.2018.22.2339

Mathematical Reviews number (MathSciNet)
MR3784523

Zentralblatt MATH identifier
06864339

Subjects
Primary: 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 20H05: Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20] 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] 30F60: Teichmüller theory [See also 32G15]
Secondary: 57M05: Fundamental group, presentations, free differential calculus 57M50: Geometric structures on low-dimensional manifolds

Keywords
pseudomodular killer intervals hyperbolic jigsaw marked ideal triangle

Citation

Lou, Beicheng; Tan, Ser Peow; Vo, Anh Duc. Hyperbolic jigsaws and families of pseudomodular groups, I. Geom. Topol. 22 (2018), no. 4, 2339--2366. doi:10.2140/gt.2018.22.2339. https://projecteuclid.org/euclid.gt/1523584824


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